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Lecture 1 Notes:
06 / 27
The first part of this class will primarily cover oscillating systems (harmonic oscillators
and waves).
These systems are very common in nature  a system displaced from equilibrium by a
small amount will tend to oscillate harmonically, unless the friction is too high for
oscillations to take place.
Mass on a spring and Hooke's Law
Simple system:
Mass on a spring.
Consider a spring that has a relaxed length
L
,
attached to a mass on a frictionless table:
Let
x
be the distance by which the spring is stretched or compressed from its relaxed
length.
It's positive if the spring is stretched, and negative if it is compressed.
If
x
is
sufficiently small (compared to
L
), we find that the force is proportional to the
displacement:
F = k x
This is Hooke's law.
The sign is negative because if the mass is displaced to the right,
the force tries to return it back to the left, towards the equilibrium position, and vice
versa.
For this reason, the force is called the
restoring force
.
The proportionality
constant is called the
spring constant
, and measures the stiffness of the spring.
Potential energy of the spring:
Since
F = dU / dx
, the potential energy is
The total energy is the kinetic energy plus the potential energy:
Systems that obey Hooke's Law are called
harmonic oscillators
.
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View Full DocumentWhy Hooke's Law?
Why does the spring (and many other systems in nature) follow such a simple force law?
There is a mathematical reason.
In general, the force
F
is a complicated function of the displacement
x
.
At the
equilibrium position,
x =
0, the force is zero.
For small displacements from equilibrium,
we can fit any
F
with a polynomial in
x
(this is known as a Taylor series, you should
have seen it in your calculus class.)
F = A + Bx + Cx
2
+Dx
3
+ .
..
Since
F(0) = 0
(equilibrium),
A =
0.
Therefore,
F = Bx + Cx
2
+Dx
3
+ .
..
Now, assume that the displacement
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 Spring '07
 Smith

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